How can I make transparant objects with a png texture
(This answer is for Blender Render, not Cycles, as you seem to have that renderer enabled in your screenshot).

First, on the texture settings panel, look for "Influence" and put a checkmark on "Alpha":



Then, back on the material settings panel, put a checkmark on "Transparency".

Finally, and this is a key step, set "Alpha" to zero here on the material transparency panel.  Think of this as a kind of minimum alpha for the material: The provided texture can make it more opaque, not less.



The results should work in the Material and Rendered viewport shading modes, and if you've selected the same image as the active image in the UV editor, it can work in the "Textured" viewport mode as well.